#PhysicsFactlet
An attempt to explain what tensors are for people with high-school Math (if you are a mathematician, this thread is not for you).
Not sure why, but tensors are often introduced in a very confused way, that makes them look more scary than they actually are.
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Let's assume you are familiar with matrices (if you aren't, chances are you don't care what a tensor is), so the fact that multiplying rows by columns a row vector with a column vector yields a scalar (i.e. a single number) should be no surprise to you.
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If we make a column of row vectors, we can repeat the process for each of them and put the results also in a column, resulting in the usual multiplication of a matrix by a vector.
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As a convention, we don't write the brackets around all the row vectors we put in a column, so our "matrix" looks a lot like a rectangular (in our case, square) array of numbers. But it is important to keep in mind that it is in fact a column of row vectors.
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As the result of multiplying a row vector with a column vector is a scalar, and the result of multiplying a matrix with a column vector is a column vector, if we multiply a matrix by both a row and a column vector, we get a scalar.
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Now, what happens if, instead of putting our row vectors in a column, we put them in a row? The usual row-by-column rule still applies, the only difference is that now the result is going to be a row vector.
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So, the weird object we created results in a row vector when multiplied by a column vector. In other words, if we want to get a scalar, we need to multiply it with two column vectors, not a row and a column vector like for a matrix.
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This is an example of a "tensor". It is a lot like a matrix, but you need to multiply it by a different number of vectors in order to get a scalar. You can build object that requires multiplication by any number of vectors you want to finally get a scalar.
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A useful way to classify them is by using two numbers to say how many row/column vectors you need to multiply them by in order to get a scalar. In this language a matrix is a (1,1) tensor, while the weird "row" thing we created would be a (0,2) tensor.
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What people do is to write them in components. The components of the vector v⃗ are labelled as vᵃ if it is a column vector, and vₐ if it is a row vector, so a row-by-column multiplication for a matrix M will look like
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To save space it is common practice not to write the summation symbol, and implicitly assume that equal indices are summed, i.e.
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There is of course a LOT more to say about tensors, but this is well beyond the scope of this already too long thread.
Point is, tensors are not the scary objects they are often depicted to be. They are a lot like matrices with a few more inputs 🙂
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#PhysicsFactlet: The "Ashcroft/Mermin Project"
I will try to (likely very slowly) go through the classic textbook "Solid State Physics" by Ashcroft and Mermin and make one or more animation/visualization per chapter. 1/
This will (hopefully) help people digest the topic and/or be useful to lecturers who are teaching about it. As with all my animations, feel free to use them.
The idea is that the animations are a companion to the book, so I will give only very brief explanations here.
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#PhysicsFactlet: The "Ashcroft/Mermin Project"
Chapter 1: The Drude Theory of metals
Electrons in a metal are accelerated by an electric field, but they keep bouncing on the metal defects/impurities. The resulting diffusion-like motion produces a roughly steady current.
#PhysicsFactlet (346)
A few days back I stumbled on a @AAPTHQ paper that has the words "Space pirates" in the tile, read it, and decided it was a lot of fun, so now you get a mini-thread and a couple of simple animations 😉 1/ aapt.scitation.org/doi/full/10.11…
Sadly the paper is not truly about space pirates, but about the "pursuit curve"problem. I.e. object A is moving, object B follows it (pointing toward A at each time), and you want the path traced by B in the pursuit.
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If A is moving at constant speed in a straight line, and B is moving at a constant but higher speed, this problem can be solved analytically (at the price of a couple of nasty integrals). 3/
#PhysicsFactlet (342) Lagrange multipliers
Strictly speaking Lagrange multipliers are not "Physics", but they are so useful to solve so many Physical problems, that it is definitively worth looking at them.
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Before we even introduce them, let's solve a super-simple problem, which will form the basis for our motivation to look into Lagrange multipliers:
Find the minimum of the function f=x²+y².
Yes, I can hear you shouting x=y=0, but let's still do the calculation.
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The way you find the minimum of a function is to check the points where all the partial derivatives are zero (in this case we have 2 variables, so we will look at the partial derivatives with respect to x and y): df/dx=2 x, df/dy=2y --> 2x=0, 2y=0 --> x=y=0.
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#PersonalOpinionOnOldishGame
You can't really finish #MonsterHunterWorld, but I have played as much as I am going to (100+ hours), so here are a few thoughts about it.
TL;DR: it is a good game with some incomprehensible flaws.
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Monster Hunter: World is the Nth (with N being a large integer) game in the the Monster Hunter series, but it was the first one I ever played (the new one, Monster Hunter: Rise is only on Nintendo Switch).
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The story is non-existent, so let's ignore it. It is just a poor excuse for you to run around some well designed maps hunting and killing dinosaur-like monsters.
There are only 5 maps in the base game, but they are large enough not to be too repetitive.
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#PhysicsFactlet (335)
Yesterday, at a small playground where my son was playing, I saw this Kugel fountain, so here comes a short thread about Kugel fountains and how they work.
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(Alt Text: a Kugel fountain slowly rotating in a sunny day.)
First of all, what is a Kugel fountain?
There are a few variations on the theme, but usually they are big stone spheres, sitting on a hemispherical hole, with water flowing from below. Despite their weight, they can spin with a small push, and keep spinning for a long time.
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How does it work?
It can't be buoyancy, as the stone sphere is a a LOT more dense than the water (we all have direct experience of stones sinking when you put them in water, and this one is not any different).
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#PhysicsFactlet (331)
"Anderson localization" is a weird phenomenon that is not well known even among Physicists, but has the habit of popping up essentially everywhere.
An introductory thread 🧵
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The idea of "localization" originally came about as an explanation (by P.W. Anderson, hence the name) of why the spins in certain materials did not relax as fast as expected. nobelprize.org/prizes/physics…
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What Anderson realized was that when you have a wave (in this case a quantum mechanical wavefunction) that propagates in a random system, interference can play a major role, and potentially impede propagation completely. journals.aps.org/pr/abstract/10…
(Paywalled)
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